Induced superconductivity in high mobility two dimensional electron gas in GaAs heterostructures
Zhong Wan, Aleksandr Kazakov, Michael J. Manfra, Loren N. Pfeiffer, Ken W. West, Leonid P. Rokhinson
IInduced superconductivity in high mobility two dimensional electron gas inGaAs heterostructures.
Zhong Wan, Aleksandr Kazakov, Michael J. Manfra,
1, 2, 3, 4
Loren N. Pfeiffer, Ken W. West, and Leonid P. Rokhinson
1, 2, 4, ∗ Department of Physics and Astronomy, Purdue University, West Lafayette, IN 47907 USA Department of Electrical Engineering, Purdue University, West Lafayette, IN 47907 USA Department of Materials Engineering, Purdue University, West Lafayette, IN 47907 USA Birck Nanotechnology Center, Purdue University, West Lafayette, IN 47907 USA Department of Electrical Engineering, Princeton University, Princeton, NJ 08544 USA
Introduction of a Josephson field effect transistor (JoFET) concept [1] sparked ac-tive research on proximity effects in semiconductors. Induced superconductivity andelectrostatic control of critical current has been demonstrated in two-dimensionalgases in InAs[2, 3], graphene[4] and topological insulators[5? –8], and in one-dimensional systems[9–11] including quantum spin Hall edges[12, 13]. Recently,interest in superconductor-semiconductor interfaces was renewed by the search forMajorana fermions[14, 15], which were predicted to reside at the interface[16–18].More exotic non-Abelian excitations, such as parafermions (fractional Majoranafermions)[19–21] or Fibonacci fermions may be formed when fractional quantumHall edge states interface with superconductivity. In this paper we develop transpar-ent superconducting contacts to high mobility two-dimensional electron gas (2DEG)in GaAs and demonstrate induced superconductivity across several microns. Super-current in a ballistic junction has been observed across 0.6 µ m of 2DEG, a regimepreviously achieved only in point contacts but essential to the formation of well sep-arated non-Abelian states. High critical fields ( > Tesla) in NbN contacts enablesinvestigation of a long-sought regime of an interplay between superconductivity andstrongly correlated states in a 2DEG at high magnetic fields[22–27].
Proximity effects in GaAs quantum wells have been intensively investigated in the past andAndreev reflection has been observed by several groups[28–31]. Unlike in InAs, where Fermi level( E F ) at the surface resides in the conduction band, in GaAs E F is pinned in the middle of thegap which results in a high Schottky barrier between a 2DEG and a superconductor and lowtransparency non-ohmic contacts. Heavy doping can move E F into the conduction band and,indeed, superconductivity has been induced in heavily-doped bulk n ++ GaAs[32]. In quantumwells similar results were obtained by annealing indium contacts[33], however the critical field ofindium is ∼
30 mT which is well below the fields where quantum Hall effect is observed.In conventional quantum well structures AlGaAs barrier between 2D electron gas (2DEG) andthe surface of the sample adds an extra 0.3 eV to the Schottky barrier when contacts are defusedfrom the top. We alleviated these problems by growing an inverted heterojunction structures,where a 2DEG resides at the GaAs/AlGaAs interface but the AlGaAs barrier with modulationdoping is placed below the 2DEG, see Fig. 1. Contacts are recessed into the top GaAs layer inorder to bring superconductor closer to the 2DEG. A thin layer of AuGe and NbN superconductorform low resistance ohmic contacts to the 2DEG after annealing. The inverted heterostructureincreases the contact area of side contacts compared to quantum well structures by utilizing allGaAs layer above the heterointerface for carrier injection (130 nm in our inverted heterostructurevs 20 −
30 nm in typical quantum wells).Here we report induced superconductivity in two devices from different wafers, sample A haslong (70 µ m) contacts separated by 1 . µ m of 2DEG, for sample B contacts are formed to the edge a r X i v : . [ c ond - m a t . m e s - h a ll ] M a r
14 25 m m A l G a A s -doping conduction band (eV) d i s t a n ce fr on s u rf ace ( n m ) G a A s R B - ( ) T (mK) (a) (c) T C . m m GaAsAlGaAs2DEG contact
FIG. 1.
Devices design and superconducting transition. (a) Scanning electron microscope (SEM)images of test devices similar to samples A and B. Enlarged region for sample B is an atomic force microscope(AFM) image of a real sample. 2D gas regions are false-color coded with green, superconducting and normalcontacts are coded with orange and blue, respectively. (b) Simulation of the conduction band energy profilein the heterostructure[34 ? ]. (c) T -dependence of resistance between contact 3 and 4 in Sample B measuredwith 10 nA ac excitation. Superconducting transition is observed at T c ≈
290 mK. of a mesa with 0 . µ m separation. Details of device fabrication are described in Methods. Whencooled down to 4 K in the dark both samples show resistance in excess of 1 MΩ. After illuminationwith red light emitting diode (LED) a 2DEG is formed and 2-terminal resistance drops to < R B − gradually decreases upon cooldown from 4 K to thebase temperature and the S-2DEG-S junctions becomes superconducting at T c ∼ . V ( I ) characteristics for two S-2DEG-S junctions (between 8-9 for sample A andbetween 3-4 for sample B) are shown in Fig. 2. Both samples show zero resistance state at smallcurrents with abrupt switching into resistive state at critical currents I c = 0 . µ A and 0 . µ Afor samples A and B respectively. V ( I ) characteristics are hysteretic most likely due to the Jouleheating in the normal state.The most attractive property of a high mobility 2DEG is large mean free path l (cid:29) ξ , with l =24 µ m and the BCS coherence length ξ = (cid:126) v f /π ∆ = 0 . µ m for sample B. Here v f = (cid:126) √ πn/m isthe Fermi velocity, n is a 2D gas density, m is an effective mass, and ∆ = 1 . k B T c = 46 µ eV is theinduced superconducting gap. Evolution of V ( I ) with T is shown Fig. 3a. Experimentally obtained T -dependence of I c is best described by the Kulik-Omelyanchuk theory for ballistic junctions( L (cid:28) l )[35], the blue curve on Fig. 3b. For comparison we also plot I c ( T ) dependence for the dirtylimit L (cid:28) √ lξ [36], which exhibits characteristic saturation of I c at low temperatures.In short ballistic junctions L (cid:28) ξ (cid:28) l the product I c (0) R N = π ∆ /e does not depend on thejunction length L . For L ∼ ξ this product is reduced by a factor 2 ξ / ( L + 2 ξ ) [37]. The measured I c R N = 83 µ V for sample B is in a good agreement with an estimate π ∆ /e · ξ / ( L + 2 ξ ) = 90 µ V.For sample A the I c R N = 19 µ V while the estimated product is ≈ µ V. The reduction is consistent -0.50 -0.25 0.00 0.25 0.50-200-1000100200-0.50 -0.25 0.00 0.25 0.50050100150 -0.50 -0.25 0.00 0.25 0.500200400600-0.50 -0.25 0.00 0.25 0.50-50-2502550 (d)(c)(a)
Current (m A) V o l t ag e ( m V ) V o l t ag e ( m V ) Ic Ir (b)
Current (m A)Current (m A) IcIr d V / d I ( ) Current (m A) d V / d I ( ) Check, different
Ic for top & bottom
Sample A Sample B
IcRn=0.244*318=77uV IcRn=0.212*82=17uV
FIG. 2.
Induced superconductivity in a high mobility 2D electron gas in GaAs.
Voltage-currentcharacteristics and differential resistance are measured between 8-9 for sample A and between 3-4 for sampleB at base temperature , dV /dI is measured with I ac = 1 nA. Induced superconductivity with zero voltageis observed with critical currents I c ∼
220 nA for sample A and I c ∼
230 for sample B. with the geometry of sample A, where a region of the 2DEG with induced superconductivity isshunted by a large region of a 2DEG in a normal state.Transparency of superconducting contacts can be estimated from the suppression of the su-perconducting gap in the S-2DEG-S junction between 3-4 in sample B. In one-dimensional junc-tions the induced gap ∆ = ∆ depends on the broadening of Andreev levels within thesemiconductor[39] Γ = (cid:126) v f L eff D D , where we introduce contacts transparencies D and D . Weassume for simplicity that D = D = 1 / (1 + Z ), where 0 < Z < ∞ is a interface barrier strengthintroduced in [38], and Bagwell’s effective channel length L eff = L + 2 ξ . Using NbN supercon-ducting gap ∆ = 2 . k B T C (NbN is a strong-coupling superconductor, T C = 11 K) we obtain Z = 0 .
2. This value is consistent with the fit of the I c vs T dependence with D as a free parameter,see Supplementary Material for details. Similar values of Z can be estimated from the analysis ofthe shape of dI/dV ( V ) characteristics at elevated temperatures, as shown in Fig. 3. At T < T c Andreev reflection at S-2DEG interfaces results in an excess current flowing through the junctionfor voltage biases within the superconducting gap ∆ /e and corresponding reduction of a differ-ential resistance dV /dI by a factor of 2. In the presence of a tunneling barrier normal reflectioncompetes with Andreev reflection and reduced excess current near zero bias, resulting in a peakin differential resistance. Within the BTK theory[38] a flat dV /dI ( V ) within ∆ /e , observed inour experiments, is expected only for contacts with very high transparency Z < .
2. For larger
Z > . I ( V ) need to be mentioned. First, we observe several sharp peaks in the resistanceat high biases (around 2 mV and 4 mV for T = 4 K). Similar sharp resonances has been observedpreviously [40], where authors attributed their appearance to the formation of Fabry-P´erot reso-nances between superconducting contacts. In our devices the superconducting region is shunted bya low resistance ( < >
10 kΩ resonances cannot be explained byresonant electron trapping between contacts. These resonances are also observed in I ( V ) character-istics of a single S-2DEG interface (measured in the S-2DEG-N configuration between contacts 3-6, I c ( m ) T (mK) -0.8 -0.4 0.0 0.4 0.80.00.51.01.52.02.5 d V / d I ( k ) Current (mA)
Updated 11/5 Δ using thouless energy in weak regime with Tc =11K Rn=2.7 kohmS-N-N plot was made from He3 dataTwo terminal between 2-20, Y axis subtracted 40 kohm from wireX not scaled -4 -2 0 2 401234 d V / d I ( k ) V(mV)11K 4K -4 -2 0 2 40.00.51.01.5 ( d V / d I ) / R N V(mV)
Z=0.1 11K 4K -4 -2 0 2 4012345678 R ( k oh m ) DCV (mV)S-N-S 20-18 4K
S-2DEG-S -4 -2 0 2 40.00.51.01.5 z=0.2 ( d V / d I ) / R N
11K 4K
V(mV) (a) (b)(c) (d)
Sample BSample B
FIG. 3.
Temperature dependence of superconductivity in a ballistic junction. (a) Evolution of theinduced superconductivity with T for sample B. The R ( I ) curves are offset proportional to T for T >
50 mK.(b) Temperature dependence of critical current I c ( T ) is extracted from (a) and compared to the expected T -dependence for ballistic and diffusive regimes (reduced I c compared to Fig. 2 is due to larger I ac = 10 nAused in this experiment). (c) High-temperature data shows Andreev reflection (excess current and reduced dV /dI around V = 0. The curves are not offset. In (d) excess current is modeled within the BTK theory[38]with Z = 0 . see Supplementary Material Fig. S2). Differential resistance does not change substantially acrossresonances, ruling out transport through a localized state. We speculate that in the contacts wherethese resonances are observed superconductivity is carried out by quasi-1D channels, and jumpsin I/V characteristics are due to flux trapping at high currents. This scenario is consistent withthe observation that peaks shift to lower currents at higher fields, see Fig. 4. The second notablefeature of our data is reduction of the zero-bias resistance by ≈ . µ m) multiple Andreev reflection is suppressed and the reduction of resistance by a factor of 2is observed, see Supplementary Material Fig. S2.Finally, we present magnetic field dependence of induced superconductivity. The low-field datais shown in Fig. 4(a,b), where black regions correspond to zero differential resistance. Inducedsuperconductivity is suppressed at ≈ . I c does not decrease to zero and abrupt jumps in I c reflect multiple flux jumps. The period of oscillations is ∼ . µ m , much smaller than the area of the 2DEG between the contacts ( ≈ µ m ). This Superconducting contact Normal contact R - T ( ) R - T ( h / e ) -0.8 -0.4 0.0 0.4 0.80.00.10.20.3 I ( m A) B ( T ) R (k ) (a) (b) (c) (d) Sample A Sample B
I ( m A) B ( m T ) R B ( - )( - ) ( h / e )
40 mK 1.7 K
B (T)2 1 2/33/5 R B ( - )( - ) ( h / e ) B(T)
Sample B Sample B
I ( m A) B ( T ) R A8-9 ( ) V V ( u V ) B/B V 𝐵/𝐵 𝜈=1
FIG. 4.
Magnetic field dependence of induced superconductivity. (a,b) Differential resistance ismeasured as a function of B and I dc for two samples at 40 mK. Induced superconductivity (black region)is observed up to 0.2 Tesla in both sample. (c) 3-terminal resistance for a sample with all normal contacts(red) and between normal and superconducting contacts in sample B [ I (2 −
4) and V (4 −
1) in Fig. 1] ismeasured at 70 mK and 40 mK respectively.
B <
B >
0) induces clockwise (counterclockwise) chiraledge channels, note resistance scales difference for two field directions. observation is consistent with the reduced I c R N product measured for this sample as discussedabove. In sample B contacts are fabricated along the edge of the mesa and 2D gas is not enclosedbetween the contacts. Consequently, I c is a smooth function of B .Competition between superconductivity and chiral quantum Hall edge states is shown in Fig. 4c,where resistance is measured in a 3-terminal configuration over a wide range of magnetic fields.Simple Landauer-Buttiker model of edge states predicts zero resistance for negative and quantizedHall resistance for positive field direction for IQHE and FQHE states, which is clearly seen in asample with all normal ohmic contacts (red curve). When a superconducting contact serves as acurrent injector (blue curve), integer ν = 1 and fractional ν = 2 / B <
0, while the same states are not quantized at proper QHE values for
B >
0. If we assumethat current injection via a superconducting contact results in an extra voltage offset at the contact V off ≈ ∆ ind /e , the measured voltage will be reduces by V off . The magenta bars for B > V − V off ) /I for V off = 140 µ V. While this offset may explain the measuredvalues for fractional states, a twice smaller V off is needed to reconcile the resistance at ν = 1.Note that induced gap is smaller at higher B . At low fields states ν = 3, 4 and 5 have resistanceminima for B < ν = 2 has a maximum. Zero resistance at ν = 1 and large resistanceat ν = 2 are in contrast to the theoretical prediction that ν = 2 state should be stronger coupledto a superconducting contact than ν = 1 [23]. Methods
The GaAs/AlGaAs inverted heterojunctions were grown by molecular beam epitaxy (MBE) onsemi-insulating (100) GaAs substrates with the heterointerface placed 130 nm below the surfaceand δ -doping layer 30-40 nm below the GaAs/AlGaAs interface. Samples were fabricated fromtwo wafers with density and mobility n = 2 . × cm − , µ = 2 × V · s/cm (sample A)and n = 1 . × cm − , µ = 4 × V · s/cm (sample B). Superconducting contacts weredefined by standard electron beam lithography. First, a 120 nm - deep trench was created bywet etching. Next, samples were dipped into HCl:H O (1 : 6) solution for 2 s and loaded into athermal evaporation chamber, where Ti/AuGe (5nm/50nm) was deposited. Finally, 70 nm of NbNwas deposited by DC magnetron sputtering in Ar/N (85% / T c = 11K and B c >
15 Tesla) with minimal strain[41]. The contacts were annealed at 500 ◦ C for 10 min ina forming gas. The measurements were performed in a dilution refrigerator with base temperature <
30 mK, high temperature data was obtained in a variable temperature He system. Sampleswere illuminated with red LED at 4 K in order to form a 2D gas, 2-terminal resistance drops from > <
500 Ω after illumination.
Acknowledgements
The work at Purdue was supported by the National Science Foundation grant DMR-1307247 (Z.W.and L.P.R.), by the Purdue Center for Topological Materials (Z.W.), and by the U.S. Departmentof Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering underAwards de-sc0008630 (A.K.) and de-sc0006671 (M.J.M.). The work at Princeton was funded bythe Gordon and Betty Moore Foundation through Grant GBMF 4420, and by the National ScienceFoundation MRSEC at the Princeton Center for Complex Materials.
Authors contribution
L.P.R. and M.J.M conceived the experiments, Z.W. fabricated samples, Z.W. and L.P.R per-formed experiments, Z.W. and L.P.R wrote the manuscript with comments from M.J.M, L.N.P.and K.W.W. designed and grew wafers, A.K. contributed to the fabrication and low temperatureexperiments.
Additional information
Supplementary information is available in the online version of the paper. Correspondence shouldbe addressed to L.P.R.
Competing financial interests
The authors declare no competing financial interests. ∗ [email protected][1] T. D. Clark, R. J. Prance, and A. D. C. Grassie, “Feasibility of hybrid Josephson field effect transistors,”Journal of Applied Physics , 2736–2743 (1980).[2] Hideaki Takayanagi and Tsuyoshi Kawakami, “Superconducting proximity effect in the native inversionlayer on InAs,” Phys. Rev. Lett. , 2449–2452 (1985).[3] Tatsushi Akazaki, Hideaki Takayanagi, Junsaku Nitta, and Takatomo Enoki, “A Josephson field effecttransistor using an InAs inserted channel In . Al . As/In . Ga . As inverted modulation-dopedstructure,” Applied Physics Letters , 418–420 (1996).[4] Hubert B. Heersche, Pablo Jarillo-Herrero, Jeroen B. Oostinga, Lieven M. K. Vandersypen, andAlberto F. Morpurgo, “Bipolar supercurrent in graphene,” Nature , 56–59 (2007).[5] Benjamin Sacepe, Jeroen B. Oostinga, Jian Li, Alberto Ubaldini, Nuno J.G. Couto, Enrico Giannini,and Alberto F. Morpurgo, “Gate-tuned normal and superconducting transport at the surface of atopological insulator,” Nat. Commun. , 575 (2011).[6] J. R. Williams, A. J. Bestwick, P. Gallagher, Seung Sae Hong, Y. Cui, Andrew S. Bleich, J. G. Analytis,I. R. Fisher, and D. Goldhaber-Gordon, “Unconventional Josephson effect in hybrid superconductor-topological insulator devices,” Physical Review Letters , 056803 (2012).[7] M. Veldhorst, M. Snelder, M. Hoek, T. Gang, V. K. Guduru, X. L. Wang, U. Zeitler, W. G. van derWiel, A. A. Golubov, H. Hilgenkamp, and A. Brinkman, “Josephson supercurrent through a topologicalinsulator surface state,” Nat. Mater. , 1–4 (2012).[8] Fanming Qu, Fan Yang, Jie Shen, Yue Ding, Jun Chen, Zhongqing Ji, Guangtong Liu, Jie Fan, Xiu-nian Jing, Changli Yang, and Li Lu, “Strong superconducting proximity effect in Pb-Bi Te hybridstructures,” Sci. Rep. , 339 (2012).[9] Yong-Joo Doh, Jorden A. van Dam, Aarnoud L. Roest, Erik P. A. M. Bakkers, Leo P. Kouwenhoven,and Silvano De Franceschi, “Tunable supercurrent through semiconductor nanowires,” Science ,272–275 (2005).[10] Pablo Jarillo-Herrero, Jorden A. van Dam, and Leo P. Kouwenhoven, “Quantum supercurrent tran-sistors in carbon nanotubes,” Nature , 953–956 (2006).[11] Jie Xiang, A. Vidan, M. Tinkham, R.M. Westervelt, and C.M. Lieber, “Ge/Si nanowire mesoscopicJosephson junctions,” Nature Nanotechnology , 208 – 213 (2006).[12] Sean Hart, Hechen Ren, Timo Wagner, Philipp Leubner, Mathias Muhlbauer, Christoph Brune, Hart-mut Buhmann, Laurens W. Molenkamp, and Amir Yacoby, “Induced superconductivity in the quantumspin hall edge,” Nat. Phys , 638–643 (2014).[13] Wenlong Yu, Yuxuan Jiang, Chao Huan, Xunchi Chen, Zhigang Jiang, Samuel D. Hawkins, John F.Klem, and Wei Pan, “Superconducting proximity effect in inverted InAs/GaSb quantum well structureswith Ta electrodes,” (2014), arXiv:1402.7282.[14] Leonid P. Rokhinson, Xinyu Liu, and Jacek K. Furdyna, “The fractional a.c. Josephson effect in asemiconductor-superconductor nanowire as a signature of Majorana particles,” Nat. Phys. , 795 – 799(2012).[15] V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P. A. M. Bakkers, and L. P. Kouwenhoven,“Signatures of Majorana fermions in hybrid superconductor-semiconductor nanowire devices,” Science , 1003 – 1007 (2012).[16] Liang Fu and C. L. Kane, “Josephson current and noise at a superconductor/quantum-spin-Hall-insulator/superconductor junction,” Phys. Rev. B , 161408 (2009).[17] Roman M. Lutchyn, Jay D. Sau, and S. Das Sarma, “Majorana fermions and a topological phasetransition in semiconductor-superconductor heterostructures,” Phys. Rev. Lett. , 077001 (2010).[18] Jason Alicea, “Majorana fermions in a tunable semiconductor device,” Phys. Rev. B , 125318 (2010).[19] David J. Clarke, Jason Alicea, and Kirill Shtengel, “Exotic non-Abelian anyons from conventionalfractional quantum Hall states,” Nat. Commun. , 1348 (2012).[20] Roger S. K. Mong, David J. Clarke, Jason Alicea, Netanel H. Lindner, Paul Fendley, Chetan Nayak,Yuval Oreg, Ady Stern, Erez Berg, Kirill Shtengel, and Matthew P. A. Fisher, “Universal topologicalquantum computation from a superconductor-Abelian quantum hall heterostructure,” Phys. Rev. X ,011036 (2014).[21] Zheng-Wei Zuo, L. Sheng, and D.Y. Xing, “Crossed Andreev reflections in superconductor and frac- up-1 tional quantum Hall liquids hybrid system,” Solid State Communications , 17 – 20 (2014).[22] A. Yu. Zyuzin, “Superconductor-normal-metal-superconductor junction in a strong magnetic field,”Phys. Rev. B , 323–329 (1994).[23] Matthew P. A. Fisher, “Cooper-pair tunneling into a quantum Hall fluid,” Phys. Rev. B , 14550–14553 (1994).[24] H. Hoppe, U. Z¨ulicke, and Gerd Sch¨on, “Andreev reflection in strong magnetic fields,” Phys. Rev.Lett. , 1804–1807 (2000).[25] Eun-Ah Kim, Smitha Vishveshwara, and Eduardo Fradkin, “Cooper-pair tunneling in junctions ofsinglet quantum Hall states and superconductors,” Phys. Rev. Lett. , 266803 (2004).[26] F. Giazotto, M. Governale, U. Z¨ulicke, and F. Beltram, “Andreev reflection and cyclotron motion atsuperconductor/normal-metal interfaces,” Phys. Rev. B , 054518 (2005).[27] J. A. M. van Ostaay, A. R. Akhmerov, and C. W. J. Beenakker, “Spin-triplet supercurrent carried byquantum Hall edge states through a Josephson junction,” Phys. Rev. B , 195441 (2011).[28] K-M. H. Lenssen, M. Matters, C. J. P. M. Harmans, J. E. Mooij, M. R. Leys, W. van der Vleuten,and J. H. Wolter, “Andreev reflection at superconducting contacts to GaAs/AlGaAs heterostructures,”Applied Physics Letters , 2079–2081 (1993).[29] T. D. Moore and D. A. Williams, “Andreev reflection at high magnetic fields,” Phys. Rev. B ,7308–7311 (1999).[30] A.A. Verevkin, N.G. Ptitsina, K.V. Smirnov, B.M. Voronov, G.N. Gol’tsman, E.M. Gershenson, andK.S. Yngvesson, “Multiple Andreev reflection in hybrid AlGaAs GaAs structures with superconductingNbN contacts,” semiconductors , 551–554 (1999).[31] Hideaki Takayanagi, Tatsushi Akazaki, Minoru Kawamura, Yuichi Harada, and Junsaku Nitta, “Su-perconducting junctions using AlGaAs/GaAs heterostructures with high h c NbN electrodes,” PhysicaE , 922 – 926 (2002).[32] Jonatan Kutchinsky, Rafael Taboryski, Claus Birger Srensen, Jrn Bindslev Hansen, and Poul Erik Lin-delof, “Experimental investigation of supercurrent enhancement in s-n-s junctions by non-equilibriuminjection into supercurrent-carrying bound Andreev states,” Physica C: Superconductivity , 4 – 10(2001).[33] A.M. Marsh, D. A. Williams, and H. Ahmed, “Supercurrent transport through a high-mobility two-dimensional electron gas,” Phys. Rev. B , 8118–8121 (1994).[34] G. L. Snider, I.-H. Tan, and E. L. Hu, “Electron states in mesa-etched one-dimensional quantum wellwires,” J. Appl. Phys. , 2849 (1990).[35] I.O. Kulik and A.N. Omel’yanchuk, “Properties of superconducting microbridges in the pure limit.”Sov. J. Low Temp. Phys. , 459 – 461 (1977).[36] I.O. Kulik and A.N. Omel’yanchuk, “Microscopic theory of the Josephson effect in superconductingbridges,” Pisma v ZETF , 216 – 19 (1975).[37] Philip F. Bagwell, “Suppression of the Josephson current through a narrow, mesoscopic, semiconductorchannel by a single impurity,” Phys. Rev. B , 12573–12586 (1992).[38] G. E. Blonder, M. Tinkham, and T. M. Klapwijk, “Transition from metallic to tunneling regimes insuperconducting microconstrictions: Excess current, charge imbalance, and supercurrent conversion,”Phys. Rev. B , 4515–4532 (1982).[39] J. D. Sau, S. Tewari, and S. Das Sarma, “Experimental and materials considerations for the topologicalsuperconducting state in electron and hole doped semiconductors: searching for non-Abelian Majoranamodes in 1d nanowires and 2d heterostructures,” Phys. Rev. B , 064512 (2012).[40] JR Gao, JP Heida, B.J. van Wees, T.M. Klapwijk, G. Borghs, and C.T. Foxon, “Superconductorscoupled with a two-dimensional electron gas in GaAs/AlGaAs and InAs/AlGaSb heterostructures,”Surface Science , 470–475 (1994).[41] D. M. Glowacka, D. J. Goldie, S. Withington, H. Muhammad, G. Yassin, and B. K. Tan, “Developmentof a NbN deposition process for superconducting quantum sensors,” (2014), arXiv:1401.2292.[42] W. Haberkorn, H. Knauer, and J. Richter, “A theoretical study of the current-phase relation inJosephson contacts,” Physics Status Solidi , K161–K164 (1978).[43] K. Neurohr, A. A. Golubov, Th. Klocke, J. Kaufmann, Th. Sch¨apers, J. Appenzeller, D. Uhlisch, A. V.Ustinov, M. Hollfelder, H. L¨uth, and A. I. Braginski, “Properties of lateral Nb contacts to a two-dimensional electron gas in an In . Ga . As/InP heterostructure,” Phys. Rev. B , 17018–17028(1996). up-2 Supplementary Materials
Superconductivity in ballistic junctions in high mobility two dimensionalelectron gas in GaAs heterostructures.
Zhong Wan, Michael Manfra, Loren Pfeiffer, Ken West, and Leonid P. Rokhinson
TEMPERATURE DEPENDENCE OF THE CRITICAL CURRENT
D 1.00 0.95 0.90 0.80 0.50 0.00 I c ( T ) / I c ( ) T (mK) data
D 1.00 0.95 0.90 0.80 0.50 0.00 I c R N ( m V ) T (mK) data 0.0 0.5 1.010152025 r m s d e v i a ti on ( a u ) transparency D c oh e r e n ce l e ng t h ( m m ) (a) (b) (c) FIG. S1.
Analysis of the temperature dependence of the critical current.
Scaled (a) and unscaled(b) product I c R N is calculated using Eq. (S1) for different transparencies D and α = 1. Red dots areexperimental data. Dashed line in (b) is for α = 0 . D = 1. In (c) root-mean-square deviation betweenthe best fit and the experimental data is shown for different D , coherence length ξ obtained from the bestfit are red triangles. Haberkorn et al. [42] generalized Kulik-Omelyanchuk current-phase relations[35, 36] to the caseof arbitrary transparency of a tunnel barrier D inserted into the Josephson junction by directlysolving Gor’kov’s equations. They obtain the following current-phase relation: I s ( φ, T ) R N = α π ∆( T )2 e sin( φ ) (cid:112) − D sin ( φ/ × tanh ∆( T )2 k B T (cid:113) − D sin ( φ/ , (S1)where ∆( T ) is the BCS gap. For α = 1 this equation interpolated between diffusive ( D = 0) andballistic ( D = 1) junctions. Critical current can be found as I c ( T ) R N = max [ I s ( φ, T ) R N ]. Weintroduce coefficient α to account for the reduction of the critical current due to the finite length ofthe junction L , α = 2 ξ/ ( L +2 ξ ) [37]. The best fit of the experimental I c R N ( T ) dependence assumingboth α and D as free parameters is obtained for D = 1 and α = 0 .
7, see Fig. S1(a,b). For the contactspacing L = 0 . µ m this α corresponds to ξ = 0 . µ m, consistent with the BCS coherence length ξ = (cid:126) v f /π ∆ = 0 . µ m. Transparency D can be related to the dimensionless barrier strength Z introduced in the Blonder-Tinkham-Klapwijk (BTK) theory[38], D = 1 / (1 + Z ), and the fit setsthe upper limit on Z , Z < .
1. The quality of the fit parameters can be assessed from Fig. S1(c),where rms error for the best fit with a fixed D and α as a free parameter (rms deviation) = (cid:80) i { [ I c ( T i ) R N ] theory − [ I c ( T i ) R N ] exp } is plotted for different D . The rms deviation has a clearglobal minimum at D →
1. Note that the coherence length for
D <
1, obtained from the fittingparameter α , becomes smaller than the estimated ξ .up-3 -4 -2 0 2 40.00.51.01.5 V(mV)
Z=1 -4 -2 0 2 40.00.51.01.5 ( d V / d I) / R N Z=0
V(mV)
11K 4K -4 -2 0 2 40.00.51.01.5 z=0.2 11K 4K
V(mV) -4 -2 0 2 40.00.51.01.5
Z=0.3
V(mV) -4 -2 0 2 40.00.51.01.5 ( d V / d I) / R N Z=0.5 11K 4K
V(mV) -4 -2 0 2 40.00.51.01.5
V(mV)
Z=0.8 -4 -2 0 2 40123456 d V / d I ( k ) V (mV)
S-2DEG-S 4K -30 -15 0 15 30101112131415 ( d V / d I) / R N
11K 4K
V(mV)
S-2DEG-N
FIG. S2.
Temperature dependence of differential resistance.
Left 6 plots: normalized differentialresistance is calculated using BKT theory, Eq. S2 for different barriers Z and temperatures between 4and 11 K with a step of 1 K. Right 2 plots: experimentally measured differential resistance between twosuperconducting contacts ( R − ) and a normal-superconducting contact ( R − ) in sample B (the normalcontact has high resistance). ANALYSIS OF EXCESS CURRENT ABOVE THE INDUCED SUPERCONDUCTIVITYGAP
Transparency of the superconductor-semiconductor interface can be estimated from the shapeof the dV /dI ( V ) characteristic, where competition between Andreev and normal reflections resultsin a peak in differential resistance when a tunneling barrier is present at the superconductor-semiconductor interface (transmission D = 1 / (1 + Z ) < dIdV ( V ) ∝ (cid:90) ∞−∞ ∂f ( E − eV ) ∂ ( eV ) [1 + A ( E ) − B ( E )] dE, (S2)where f ( E ) is the Fermi Dirac function and A ( E ) and B ( E ) are energy-dependent Andreev andnormal reflection coefficients, respectively. Both coefficients depend on the gap of NbN ∆ = ∆( T )with T c = 11 K and the interface barrier strength Z . In Fig. S2 we plot differential resistancefor different values of Z . At low T for Z = 0 the barrier is transparent ( D = 1) and all incidentelectrons are Andreev reflected, which leads to the a reduction of differential resistance by a factorof 2 within the energy gap ∆ . When Z is finite, part of the incident electrons undergoes normalreflection which results in the increase of the resistance within the gap.The exact shape of experimental curves differ from the shape predicted by the BKT theory, themost important deviation being sharp minima near V = 0 observed at T close to T c as comparedto a much smoother BKT dependence. To account for a similar sharpening of a zero-bias peakin less transparent contacts ( Z >
2) it has been assumed that a thin normal region is formedbetween NbN contacts and a 2DEG[43]. This more elaborate theory introduces two more fittingparameters for the superconducting-normal and normal-2DEG interfaces, but does not change themain qualitative prediction of a simpler BTK theory: appearance of a peak near V = 0 for Z > . dV /dI ( V ) characteristics.up-4Experimentally, we observe no zero-bias peak in dV /dI ( V ) characteristics measured betweentwo superconducting contacts R − (S-2DEG-S) or between superconducting and normal contacts R − (S-2DEG-N), see Fig. 3 and S2, thus we can set an upper limit Z < . D > .
96 for our contacts.
COMPARISON BETWEEN CONVENTIONAL HETEROSTRUCTURES ANDINVERTED SINGLE INTERFACE HETEROSTRUCTURE
Comparison between conventional heterostructures and inverted heterostructures used in thiswork is shown in Fig. S3. In conventional quantum well (b,d) and single interface heterostructuresAlGaAs barrier between 2D gas and the surface adds 0.3 eV to the Schottky barrier if contactsare defused from the surface. For side contacts inverted single heterointerface (a) increases theexposed GaAs cross section for Cooper pair injection.
AlGaAs aAlGaAs AlGaAs C ondu c t i on band ( e V ) bAlGaAs cAlGaAsAlGaAs distance from surface(nm) d FIG. S3.